Self-inductance ($\Lambda =LI,{\mathcal{E}}_{\mathrm{ind}}=-L\dot{I}$)

Self-inductance is inductance associated with a coil's own electric current, rather than magnetic flux supplied by another coil. In the linear model, fixed geometry and magnetic response give constant $L$ in $\Lambda =LI$. With the turn orientations defined in magnetic flux linkage, Faraday's law of induction gives the self-induced electromotive forceThe final equality requires constant $L$; if the geometry changes, use $-d\Lambda /dt$. The minus sign expresses Lenz's law: the induced electromotive force opposes the change in electric current. It does not always oppose the electric current itself. When positive electric current decreases, the induced electromotive force acts in its direction. ^[ref].

For terminal sign bookkeeping, label the ideal inductor's terminals $a,b$, choose positive electric current entering $a$ and leaving $b$, and define $v_L$ as the terminal electric potential difference drop from $a$ to $b$ in the lumped electrical circuit model. This passive convention givesThus a coil with self-inductance $0.5\mathrm{H}$ and increasing electric current $dI/dt=12A/s$ has terminal drop $v_L=6\mathrm{V}$ and induced electromotive force ${\mathcal{E}}_{\mathrm{ind}}=-6\mathrm{V}$. These signs describe the same process with different reference conventions. The RL circuit uses the terminal-drop convention throughout.

A finite terminal electric potential difference cannot make the electric current of an ideal inductor with constant $L>0$ jump. Integrating its differential equation givesIf $v_L$ stays bounded, the integral tends to zero as the interval shrinks, ensuring continuity of electric current. A step in terminal electric potential difference changes $dI/dt$ immediately, while electric current remains continuous. At constant electric current, the self-induced electromotive force is zero, although the inductor's magnetic energy can remain nonzero.

Two long round wires of radius $a_w\ll d$ carrying opposite electric currents form a loop with the displayed leading external self-inductance per unit length. Integrating their summed magnetic field across a connecting ribbon gives the logarithm; magnetic-energy integration gives the same leading value. The zero-radius limit diverges. Uniform direct current inside two nonmagnetic round wires contributes another ${\mu }_0/(4\pi )$, while an ideal surface current has no interior contribution.

An inductor is an electrical circuit component designed to provide self-inductance, often a coil such as a solenoid. An ideal linear inductor has constant $L>0$ and zero electrical resistance. Its stored magnetic energy, with zero chosen at zero electric current, isFor self-inductance $0.5\mathrm{H}$ and electric current $3\mathrm{A}$, this gives $U=2.25\mathrm{J}$. Unlike Joule heating, ideal storage is recoverable: the inductor can return energy to an electrical circuit when its electric current decreases. ^[ref].

The short derivation uses the terminal convention in self-inductance. The power absorbed by an ideal inductor isStarting from zero electric current, the integral of this power givesPositive $P_L$ increases stored magnetic energy; negative $P_L$ returns energy. At constant electric current $3\mathrm{A}$ in the example, $P_L=0$ and induced electromotive force is zero, but $U=2.25\mathrm{J}$ still resides in the magnetic field. The battery in the steady RL circuit supplies Joule heating in the electrical resistance, rather than further increasing the inductor's magnetic energy.

During the closed-loop release in RL circuit, stored magnetic energy plus accumulated Joule heating stays $2.25\mathrm{J}$. A real coil also has electrical resistance, so its Joule heating must be included separately from ideal inductor storage. The constant-$L$ formulas assume a fixed linear magnetic response.

Following a change in applied electromotive force, a series inductor and resistor obey $L\dot{I}+RI=\mathcal{E}(t)$. For a constant applied electromotive force, the difference from the final electric current decays as $e^{-tR/L}$. Neglecting self-inductance is justified only when the forcing varies slowly compared with the time constant $L/R$, or when considering a steady state.

For an ideal linear inductor, $V_L=L{\dot{I}}_L$. Integrating shows that voltage with no impulse preserves current continuity at switching. This supplies an initial condition independently of capacitor voltage continuity; the two state variables must not be replaced by an assumed zero current derivative.

An ideal voltage source $V_0$ and series resistor $R_1$ feed parallel branches $L$ and $R_2$. Thévenin's theorem gives $R_{\mathrm{th}}=R_1{R}_2/(R_1+R_2)$ and $V_{\mathrm{th}}=V_0{R}_2/(R_1+R_2)$. Starting with zero inductor current, $i_L=(V_0/R_1)(1-e^{-t/\tau })$ and $i_2=V_0{e}^{-t/\tau }/(R_1+R_2)$. Switch-on heat in $R_2$ is $L{V}_0^2/[2R_1(R_1+R_2)]$. Disconnecting the source leaves the stored magnetic energy $L{i}_L^2/2$ to dissipate through $R_2$, with time constant $L/R_2$.

For a reversible device with flux linkage $\Psi$, electrical power is $I\dot{\Psi }$, so stored energy changes by $dU=Id\Psi$. This reduces to $U=L{I}^2/2$ only for a fixed linear inductance. Hysteresis makes energy path dependent and introduces loss.